MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 281 
histories and a collection of family data drawn from private pedigrees and other 
sources. In the case of the daughter, no marriage was taken which had not lasted 
at least 15 years, or until the death of husband or wife. In the case of the mother 
no limitation whatever was made, the number of brothers and sisters of the daughter, 
including herself, being counted. Weight was given to the fertility of the mother, 
for every possible case that could be got from the records under the above conditions 
was extracted. It is quite possible that a certain proportion of offspring dying in 
early infancy have not been entered in the records. 
If M;„, be the mean fertilities of mother and daughter, o-,„, a-j, their standard 
deviations, and their correlation, we found : 
= 3-494, = 6-225, 
CTa — 2-975, (r„j = 3-052, 
= -0418. 
Clearly owing to the near equality of and cr„i the regression of daughter’s on 
mother’s fertility is sensibly equal to the correlation. 
The probable error of is determined by the formula given by Pearson and 
Filon"' to be -0101, or is four times its probable error. 
We thus conclude : 
(i.) That fertility is inherited in the female line. 
(ii.) That its effects are very largely screened by the factors to which we have 
previously referred. 
Had we started with no limitation as to the daughter’s family, it is highly probable 
that would scarcely have been sensible relatively to its probable error, and, 
therefore, small series without due regard to screening causes may easily lead the 
recorder to suppose that fertility is not inherited. 
Supposing we exclude from the daughters the 775 barren marriages, we find the 
mean for 3643 cases of fertile marriages to be 5-237. Comparing this fertility with 
the observed fertility 6-225 of mothers, a superficial inquirer miglit at once consider 
that a diminution of fertility has taken place. The fact is that neither of the results, 
or gives the actual fertility of the mothers or daughters. These are the 
means M"i and of formulse (viii.) and (vii.) of the theoretical investigation. 
Let us ajDply the theory developed to our statistics. In the first place we note 
that r is small ; hence is still smaller, and thus by (xv.) cr'b will not differ much 
from o-o. Since cr"i will be generally less than 0-1 by (xvi.), it follows that a \ will 
probably be less than cr^. Approximately, we can take ao = 3. Turning to (vii.) we 
see that M'b cannot, since r is small, differ widely from Mo. If there be no secular 
* “Contributions to Theoi’y of Evolution.—IV.’’ ‘Phil. Trans,,’ A, vol. 191, p. 242. 
VOL. CXCII.—A. 2 O 
